The spin-up from rest of a contained homogeneous free-surface fluid has been examined in the laboratory for a variety of non-axisymmetric containers. It was found that in the spin-up process three stages can be distinguished before the fluid reaches the ultimate state of rigid-body rotation. When the container starts spinning, the non-axisymmetric lateral tank boundaries induce horizontal pressure gradients, and as a result relative flows arise instantaneously after the start of the experiment. The absolute vorticity of the starting flow is zero, and a description can be given in terms of potential theory. Theoretical solutions have been derived for a number of geometries, and comparison with experimentally observed streamline patterns shows good agreement. In the next stage, flow separation sets in, in most cases leading to locally intense three-dimensional turbulent flows. The basic rotation causes a transition from three-dimensional to two-dimensional motion, and a subsequent organization of the relative flow into a number of cells is observed. During the final stage, the flow in these cells gradually decays owing to the spin-up/spin-down mechanism provided by the Ekman layer at the bottom of each cell, until eventually the fluid is in solid-body rotation.